In this work, we answer the fundamental question of identifying the fastest possible global pulse for a CZ gate and for its three qubit generalization, the C\(_2\) Z gate, satisfying the requirements (i)-(iii) above. While we assume the basic level structure of Ref. [1]} (see Fig. REF (a)), our results provide original pulse schemes. Key results include: (i) For the CZ gate, we find that the time-optimal global pulse corresponds to a pulse where the laser amplitude is kept constant, while the laser phase is varied continuously and smoothly in time. The resulting pulse is about 10% faster than that of Ref. [2]} – which we thus find to be already an excellent pulse. Interestingly, the found global CZ gate remains time-optimal even when considering pulses making use of single-site addressability, demonstrating that the latter is not necessary for two-qubit operations.
(ii) For the C\(_2\) Z gate, we find two qualitatively different time-optimal global pulses, which differ by less than 1% in speed. Interestingly, both pulses are even faster than the pulse proposed in Ref. [3]}, which requires single site addressability. To our knowledge this is the first work to identify time-optimal pulses for C\(_2\) Z gates. (iii)
We demonstrate that the found time-optimal pulses can be adjusted to minimize errors arising from the decay of the Rydberg state or imperfect blockade. (iv) All pulses for CZ and C\(_2\) Z gates correspond to smooth time evolutions of the laser phase that can be fully described by just a few variational parameters and (v) allow for reaching theoretical fidelities compatible with most error correction schemes.
